To determine if the given three vectors in $\mathbb{R}^4$ are linearly independent, we need to analyze if there exists a nontrivial linear combination that results in the zero vector. Specifically, we have three vectors:

$\mathbf{v_1} = \begin{bmatrix} 3 \\ 2 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} -1 \\ 2 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v_3} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ -1 \end{bmatrix}$

The process involves forming a matrix $A$ with these vectors as columns and then performing row reduction to see if the matrix has a pivot in every column. If it does, the vectors are linearly independent; otherwise, they are dependent.

The matrix $A$ is:

$A = \begin{bmatrix} 3 & -1 & 1 \\ 2 & 2 & 2 \\ 1 & 5 & 3 \\ 1 & -3 & -1 \end{bmatrix}$

I'll proceed with row reduction to determine if there are pivots in each column.

1. Row Reduction: Apply elementary row operations to transform $A$ into row echelon form.
2. Result Analysis: If each column has a leading 1 (pivot), the vectors are independent. If not, they are dependent.

Let me perform these calculations to verify the result.The rank of the matrix $A$ is 2, which means that there are only two linearly independent columns. Since we have three columns but the rank is less than 3, this indicates that the three vectors are linearly dependent in $\mathbb{R}^4$.

Would you like further explanation on linear dependence or how to interpret the rank in this context?

Here are five related questions you might find helpful:

1. How do we interpret the rank of a matrix in terms of vector independence?
2. What is the geometric meaning of linear dependence in $\mathbb{R}^4$?
3. Can we determine dependence or independence of more vectors than the space dimension?
4. How does row reduction help in identifying vector independence?
5. What does it mean if the rank of a matrix equals the number of columns?

Tip: Remember, if the rank of a matrix is equal to the number of columns, the columns (or vectors) are linearly independent.